Coupling the valley degree of freedom to antiferromagnetic order

Abstract

Conventional electronics are based invariably on the intrinsic degrees of freedom of an electron, namely its charge and spin. The exploration of novel electronic degrees of freedom has important implications in both basic quantum physics and advanced information technology. Valley, as a new electronic degree of freedom, has received considerable attention in recent years. In this paper, we develop the theory of spin and valley physics of an antiferromagnetic honeycomb lattice. We show that by coupling the valley degree of freedom to antiferromagnetic order, there is an emergent electronic degree of freedom characterized by the product of spin and valley indices, which leads to spin-valley-dependent optical selection rule and Berry curvature-induced topological quantum transport. These properties will enable optical polarization in the spin-valley space, and electrical detection/manipulation through the induced spin, valley, and charge fluxes. The domain walls of an antiferromagnetic honeycomb lattice harbors valley-protected edge states that support spin-dependent transport. Finally, we use first-principles calculations to show that the proposed optoelectronic properties may be realized in antiferromagnetic manganese chalcogenophosphates (MnPX3, X = S, Se) in monolayer form.

Fig. 1.

The antiferromagnetic honeycomb lattice. (A) Néel afm of a honeycomb lattice. ↑ and ↓, staggered spin-density wave. (B) Spin-dependent lattice potential corresponding to the afm order. (C) Low-energy quasiparticle bands of an NN hopping afm Hamiltonian. Solid lines assume zero spin–valley coupling, whereas dashed lines take into account the spin–valley coupling. With the hopping parameter t as the energy unit, the mass and the spin–valley coupling parameter are set to m = 0.4t and δ = 0.2t, respectively. The zero of energy is set to midgap, the Dirac point energy when m = 0 and δ = 0. (Inset) Brillouin zone and high-symmetry points.

Fig. 2.

Spin and valley physics of an afm honeycomb lattice. (A) s ⋅ τ–selective CD in the absence of spin–orbit coupling. (B) When spin–valley coupling is present, the valleys can be doped asymmetrically. (C) Electron spin (bottom of conduction bands) fluxes under the action of Berry curvature of the Bloch bands and in-plane electric field. The spin-up and spin-down currents are shown in blue and green, respectively. Solid and dashed lines stand for the currents from K₊ and K₋, respectively. The spin and valley, (s, τ), indices are indicated in parentheses. E is an applied in-plane electric field.

Fig. 3.

Spin-polarized edge states. (A) Band structure in the presence of a zigzag magnetic domain on the afm honeycomb lattice, derived from NN hopping Hamiltonian. A model of a zigzag domain wall is used, which extends 50 unit cells on both sides. The bulk states are lumped into shaded blobs. With the hopping parameter t as the energy unit, the mass is set to m = t/2. The edge states of two spins are shown in blue and green. (B) Wavefunctions of two spins at the same Dirac points. The amplitudes are convolved on 2D Gaussians centered on lattice sites for visualization (scale bar in arbitrary units).

Fig. 4.

Structure and afm of monolayer MnPX₃. (A) Structure showing the unit cell. Purple spheres are Mn, yellow X, and gray P. Some of the computed bond lengths are P–P = 2.22 Å, P–S = 2.04 Å for MnPS₃, and P–P = 2.24 Å, P–Se = 2.22 Å for MnPSe₃. (B) Spin densities in one unit cell, presenting the antiferromagnetic configuration (the isosurface of 0.4 e/ų). (Left) up-spin; (Right) down-spin.

Fig. 5.

Electronic structure of manganese chalcogenophosphates from density functional theory (DFT) calculations. (A) The band structures of MnPX₃ (X = S, Se) near the band gaps. The energy scales are zeroed to the Fermi level. (B) The momentum-resolved degrees of circular polarization of MnPX₃ (X = S, Se), η^(s)(k), between the top of valence bands and the bottom of conduction bands. Only the values of one spin are presented, as in our calculations the other spin takes values equal in magnitude but with opposite signs over the Brillouin zone. At Γ, the computed optical selectivity is nonzero. This is a numerical artifact because of indeterminacy in η^(s)(k) in the presence of band degeneracy (apart from the spin degeneracy) in the valence bands (A).